Rabbits, triangles, and triplets. These three things are linked by an important series often characterised by shells and flowers. But what do the Fibonacci numbers (and their subsequent sequence) have to offer the world of finance?

In mathematics, the Fibonacci series refers to the ordered sequence of numbers described by Leonardo of Pisa, a 12th-century Italian mathematician.

\(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …\)

Each element in the series is known as a Fibonacci number.

The history of the Fibonacci sequence

This sequence was described by Fibonacci as the solution to a rabbit breeding problem: “a certain man has a pair of rabbits in a closed space and wants to know how many are created from this pair in a year when, according to nature, each couple requires one month to grow old and each subsequent month procreates another couple.” (Laurence Sigler, Fibonacci’s Liber Abaci, pg. 404).

The answer to this question is as follows:

• 1st month: we start from a pair of rabbits.
• 2nd month: the couple grows older but does not procreate.
• 3rd month: the pair procreates another pair (that is, we now have two couples).
• 4th month: the first couple procreate and the second age without procreating (we now have three couples).
• 5th month: the two older couples procreate, while the new pair ages (five pairs in total).
• Etc.

Schematically, this would be:

Where:

• Black arrow: the pair of rabbits ages.
• Red arrow: the pair of rabbits age for the first time (and therefore don’t procreate).
• Green arrow: the pair of rabbits procreates.

How are the Fibonacci numbers calculated?

There are different ways to calculate Fibonacci numbers:

1. From the numbers \(0\) and \(1\), the Fibonacci numbers are defined by the function:
   $$
   \begin{align}
   f_{n}&=f_{n-1} +f_{n-2}\\
   f_0&=0\\
   f_1&=1\\
   f_{2}&=f_{1}+f_{0}=1\\
   f_{3}&=f_{2}+f_{1}=2\\
   …
   \end{align}
   $$
2. A generating function for any sequence \(a_{0},a_{1},a_{2},…\) is the function \(f(x)=a_{o}+a_{1}x+a_{2}x^{2}+…\), that is, a formal power series where each coefficient is an element of the sequence. Fibonacci numbers have the generating function:
   $$f(x) = \frac{x}{1-x-x^{2}}$$
3. Explicit formula, this way of calculating Fibonacci numbers uses the golden number expression:
   $$f_{n}=\frac{\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(1-\frac{1+\sqrt{5}}{2}\right)^{n}}{\sqrt{5}}=\frac{\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}}{\sqrt{5}}$$

Fibonacci numbers in Mathematics

Golden numbers

The golden number, gold number or divine proportion, is the numerical value of the proportion held by two segments of line \(a\) and \(b\) (with \(a\) longer than \(b\)): the total length is to segment \(a\), as \(a\) is to segment \(b\).

One property stands out among many: the number itself, its square and its inverse, have the same decimal figures:

$$
\begin{align}
\phi&=1.\color{red}{6180339887}\ldots\\
\phi^2&=\phi+1=2.\color{red}{6180339887}\ldots\\
\frac{1}{\phi}&=\phi-1=0.\color{red}{6180339887}\ldots\\
\end{align}
$$

The ratio or quotient between Fibonacci terms and the immediately preceding one varies continuously, but stabilizes in the golden number:

$$
\lim_{n \rightarrow \infty}\frac{f_{n}+1}{f_{n}}=\phi\approx1.6180339887
$$

Pascal’s triangle

Pascal’s triangle is a representation of the binomial coefficients ordered in a triangle form. That is, each row of the triangle represents the coefficients of the monomials that appear in the development of the binomial \((a+b)^{n}\) (taking the top \(1\) as the power \(n=0\)) or, in the same way, the coefficients appear in Newton’s binomial coincide with the elements appearing in each row of the Pascal triangle.

The triangle’s construction is as follows:

We put a \(1\) in the triangle’s top vertex. Then, in the next row, we place a \(1\) on the right and a \(1\) on the left. In the lower rows, place 1st at the ends and for others, the sum of the numbers directly above on either side.

This triangle has a number of curious properties:

1. If we add the elements of each row, we get the powers of \(2: 1, 2, 4, 8, 16,\) etc.
2. Adding two consecutive elements of the diagonal \(1-3-6-10-15,\) etc., we get a perfect square: \(1, 4, 9, 16, 25,\) etc.
3. If the first number in a row (after \(1\)) is a prime number, then all other numbers are divisible by that prime number (excluding the 1s). For example, in row \(1-7-21-35-35-32-7\), the first number is \(7\), which is prime. The rest \((7,21,35)\) are all divisible by \(7\).

But the main curiosity is the property relating to the Fibonacci numbers:

Pythagorean triples

A Pythagorean triple consists of three elements (\(a, b, c\)) that satisfy \(a^{2}+b^{2}=c^{2}\) (Pythagorean theorem).

There’s a close relationship between the Fibonacci numbers and the Pythagorean triples. If we take four consecutive numbers from the Fibonacci sequence, \((w, x, y, z)\), we can get a Pythagorean triple if we make the following assignments:

1.    Let \(a\) be the product of the numbers belonging to the extremes \(a = xz\).
2.    Let \(b\) be the double of the product of the intermediate numbers \(c = 2yw\).
3.    Let \(c\) be the sum of the product of the odd numbers and the product of the even numbers \(c = xw + zy\).

Therefore (\(a, b, c\)) is a Pythagorean triple.

Fibonacci numbers in trading techniques

In trading, Fibonacci numbers appear in so-called Fibonacci studies. Fibonacci studies encompass a series of analysis tools based on sequence and Fibonacci ratios, which represent geometric laws of nature and human behaviour applied to financial markets.

The most popular of these tools are Fibonacci retracements, extensions, arcs, fan and time zones. Other tools include the Fibonacci eclipse, spiral and canals. 

If you want to know how some of these tools work in financial markets, read our post “Fibonacci retracement and extensions“.

Read this post in Spanish.